# Classifying Causal Structures: Ascertaining when Classical Correlations are Constrained by Inequalities

arXiv: https://arxiv.org/abs/2308.02380 · HTML: https://mlsanity.com/paper/2308.02380/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 50 references that are in the map.
- First posted on arXiv 4 Aug 2023.

## Builds on (most cited first, 7 of 7)

- [Causal Networks: Semantics and Expressiveness](https://mlsanity.com/paper/1304.2379/) (2013)
- [On the Testability of Causal Models with Latent and Instrumental Variables](https://mlsanity.com/paper/1302.4976/) (2013)
- [On the Logic of Causal Models](https://mlsanity.com/paper/1304.2355/) (2013)
- [Separators and Adjustment Sets in Causal Graphs: Complete Criteria and an Algorithmic Framework](https://mlsanity.com/paper/1803.00116/) (2018)
- [The Inflation Technique for Causal Inference with Latent Variables](https://mlsanity.com/paper/1609.00672/) (2016)
- [Beyond Bell's Theorem II: Scenarios with arbitrary causal structure](https://mlsanity.com/paper/1404.4812/) (2014)
- [Entropic Inequality Constraints from e-separation Relations in Directed Acyclic Graphs with Hidden Variables](https://mlsanity.com/paper/2107.07087/) (2021)

## Abstract (arXiv)

The classical causal relations between a set of variables, some observed and some latent, can induce both equality constraints (typically conditional independences) as well as inequality constraints (Instrumental and Bell inequalities being prototypical examples) on their compatible distribution over the observed variables. Enumerating a causal structure's implied inequality constraints is generally far more difficult than enumerating its equalities. Furthermore, only inequality constraints ever admit violation by quantum correlations. For both those reasons, it is important to classify causal scenarios into those which impose inequality constraints versus those which do not. Here we develop methods for detecting such scenarios by appealing to d-separation, e-separation, and incompatible supports. Many (perhaps all?) scenarios with exclusively equality constraints can be detected via a condition articulated by Henson, Lal and Pusey (HLP). Considering all scenarios with up to 4 observed variables, which number in the thousands, we are able to resolve all but three causal scenarios, providing evidence that the HLP condition is, in fact, exhaustive.